Theorem 2euswapv 2714

Description: A condition allowing to swap an existential quantifier and a unique existential quantifier. Version of 2euswap 2729 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 10-Apr-2004.) (Revised by Gino Giotto, 22-Aug-2023.)

Assertion

Ref	Expression
2euswapv	⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2euswapv

Step	Hyp	Ref	Expression
1		excomim 2169	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
2	1	a1i 11	. . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑))
3		2moswapv 2713	. . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))
4	2, 3	anim12d 610	. 2 ⊢ (∀𝑥∃*𝑦𝜑 → ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑)))
5		df-eu 2653	. 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑))
6		df-eu 2653	. 2 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑))
7	4, 5, 6	3imtr4g 298	1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1534  ∃wex 1779  ∃*wmo 2619  ∃!weu 2652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653

This theorem is referenced by:  2eu1v  2735  euxfr2w  3707  2reuswap  3733  2reuswap2  3734  reuxfrdf  30253